Only way that strat is viable, is having Grasp - else you get outtraded/go even. This kind of trade also goes both ways, as Jax can do the exact same thing. After 6, you're gonna start regretting having Grasp on Darius.
If Jax reads you right, or flat out decides to do the same to you, he wins the trade by basically ignoring your W and AAs with E, while getting burst on ya - and in worst case scenario, where he gets hit by a Q after E, and retreats, he goes even at worst. What's even better, following the E/Q trade, Jax has Q to either keep going, if he thinks he can win an extended trade or back off if he thinks he can't.
And who wins a fight if both are at half health? Jax, because he gets dmg off faster.
This is all leads back to my point, that Jax holds the cards and tools to beat Darius, and at worst, go even. As I stated, if both play 100% correct, Jax will win in the end, due to shorter CD trade power, that's stronger than Darius' burstwise.
Only way Darius wins, is if Jax fucks up, and eats a full (AA)-W-E-Q combo, without having done responded - there's no way he wins a trade after that, and can only retreat - but guess what? It all depends on him being bad, not Darius being good.
Now that's out of the way, out of lane, let's say Darius was the actually better player and went 2/0 in lane, and got a 50 cs lead - once Jax has SoS, he's back in the game. I watched Ravenborn alot, when trying to get better at Darius - he beat a jax hard in lane, to a point he had a full Deadmans on the Jax, (TF/Steraks/DP) and Jax (TF/SoS) actually won 1v1, because of that god forsaken shit item. He couldn't run, he couldn't do dmg to him (couldn't hit Jax with Q or AAs/W, kek) - it was URF vs non URF...
And to re-re-reiterate - concidering a competent jungler, it's no brainer who should win. All it takes is a one kill lead for Jax, and he can literally run Darius down - the other way around, Jax can always escape.
Shit could be solved by actually letting these cancer champions have actual CDs in the early game, give conq a time stack kinda like Atmas (conq stacks 1 every sec, for every AA/ability to up to 5 stacks), and removing that SoS passive.
The only situation you're right, where Darius wins, is a lvl 3+ iron slugfest at full health, where both hit all their abillities and AAs, then Darius should win - as 5 stack Darius IS able to put out alot of dmg... It's just not how real life goes. ;)